Problem 72 - Algebra 2

Question

Graph \(f(x)=3 x^{2}-1 .\) Translate the graph right five units and down two units. What is the vertex of the new graph?

Step-by-Step Solution

Answer

The vertex of the new graph is (5, -3).

Step 1: Understand the original function

The given function is a quadratic equation in the form of \(f(x) = ax^{2} + bx + c\), where \(a=3\), \(b=0\), and \(c=-1\). The vertex form of a quadratic function is \(f(x) = a(x-h)^{2} + k\), where \((h, k)\) is the vertex of the parabola. Since the original function has no \(x\)-linear term, its vertex is at \((0, -1)\) since \(h = 0\) and \(k = c = -1\).

Step 2: Translate the graph right

To translate the graph of the function to the right by 5 units, you modify the \(h\) value in the vertex form. The new function after translating to the right by 5 units is \(f(x) = 3(x-5)^{2}-1\), because you subtract 5 from \(x\) to move the graph to the right.

Step 3: Translate the graph down

To translate the graph down by 2 units, you modify the \(k\) value in the vertex form. The new function after translating down by 2 units is \(f(x) = 3(x-5)^{2} - 1 - 2 = 3(x-5)^{2} - 3\), because you subtract 2 more from the \(k\) value, moving the vertex down.

Step 4: Determine the new vertex

After the transformations, the new vertex form of the function is \(f(x) = 3(x-5)^{2} - 3\). The new vertex \((h, k)\) can be read directly from the function as \((5, -3)\). This is the vertex of the graph after being translated right by 5 units and down by 2 units.

Key Concepts

Quadratic Functions

Quadratic functions are fundamental elements of algebraic studies and form parabolic graphs when plotted. They are generally expressed in the standard form, which is \(f(x) = ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants and \(a \eq 0\). The term \(ax^2\) defines the curvature of the parabola, \(bx\) shifts the parabola left or right and affects its width, and \(c\) moves the parabola up or down on the coordinate plane. In our exercise, \(f(x)=3x^2 -1\), shows a parabola that opens upwards due to the positive coefficient of \(x^2\). Knowing the nature of quadratic functions is essential in predicting the graph's shape and understanding the impact of transformations such as translations.

Vertex Form of a Parabola

The vertex form of a parabola is an insightful way to express a quadratic function. It is written as \(f(x) = a(x-h)^2 + k\), where the vertex of the parabola is located at the point \(h, k\). It's important to note that \(h\) and \(k\) represent the horizontal and vertical shifts from the origin, respectively. One significant advantage of this form is that it clearly shows the vertex, making it easier to graph the parabola and apply transformations. In our context, the original function is \(f(x)=3x^2-1\), which can be thought of as \(f(x)=3(x-0)^2-1\), indicating the vertex at point \(0, -1\).

Translating Graphs

Graph translation is a form of transformation that shifts a graph horizontally, vertically, or both without changing its shape or orientation. To translate a graph horizontally, we adjust the \(x\) values, and to translate it vertically, we modify the \(y\) values. For a quadratic function in vertex form, \(f(x) = a(x-h)^2 + k\), translating the graph right by \(p\) units would involve replacing \(x\) with \(x-p\), thus \(h\) becomes \(h+p\). To translate it down by \(q\) units, we would subtract \(q\) from \(k\), changing the vertex form to \(f(x) = a(x-h)^2 + (k-q)\). In our exercise, translating the graph right by 5 units and down by 2 units results in a new vertex at point \(5, -3\), shifting our original vertex from \(0, -1\) to this new location. This translation helps visualize the positional changes on the graph due to the applied transformations.